Theorem subgmulg 13394

Description: A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)

Hypotheses

Ref	Expression
subgmulgcl.t	|- .x. = (.g ` G )
subgmulg.h	|- H = ( G ↾s S )
subgmulg.t	|- .xb = (.g ` H )

Assertion

Ref	Expression
subgmulg	|- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) = ( N .xb X ) )

Proof of Theorem subgmulg

Step	Hyp	Ref	Expression
1		subgmulg.h	. . . . . 6 |- H = ( G ↾s S )
2		eqid 2196	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
3	1, 2	subg0 13386	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
4	3	3ad2ant1 1020	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( 0g ` G ) = ( 0g ` H ) )
5	4	ifeq1d 3579	. . 3 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
6	1	a1i 9	. . . . . . . . . . 11 |- ( S e. (SubGrp ` G ) -> H = ( G ↾s S ) )
7		eqid 2196	. . . . . . . . . . . 12 |- ( +g ` G ) = ( +g ` G )
8	7	a1i 9	. . . . . . . . . . 11 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
9		id 19	. . . . . . . . . . 11 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
10		subgrcl 13385	. . . . . . . . . . 11 |- ( S e. (SubGrp ` G ) -> G e. Grp )
11	6, 8, 9, 10	ressplusgd 12831	. . . . . . . . . 10 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
12	11	3ad2ant1 1020	. . . . . . . . 9 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( +g ` G ) = ( +g ` H ) )
13	12	seqeq2d 10563	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) )
14	13	adantr 276	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) )
15	14	fveq1d 5563	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) = ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) )
16	15	ifeq1d 3579	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) = if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) )
17		simprl 529	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> -. N = 0 )
18		simprr 531	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> -. 0 < N )
19		simp2 1000	. . . . . . . . . . . 12 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> N e. ZZ )
20		ztri3or0 9385	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) )
21	19, 20	syl 14	. . . . . . . . . . 11 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N < 0 \/ N = 0 \/ 0 < N ) )
22	21	adantr 276	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( N < 0 \/ N = 0 \/ 0 < N ) )
23	17, 18, 22	ecase23d 1361	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> N < 0 )
24		simpl1 1002	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> S e. (SubGrp ` G ) )
25	19	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> N e. ZZ )
26	25	znegcld 9467	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> -u N e. ZZ )
27	19	zred 9465	. . . . . . . . . . . . . . 15 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> N e. RR )
28	27	lt0neg1d 8559	. . . . . . . . . . . . . 14 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N < 0 <-> 0 < -u N ) )
29	28	biimpa 296	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> 0 < -u N )
30		elnnz 9353	. . . . . . . . . . . . 13 |- ( -u N e. NN <-> ( -u N e. ZZ /\ 0 < -u N ) )
31	26, 29, 30	sylanbrc 417	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> -u N e. NN )
32		eqid 2196	. . . . . . . . . . . . . . . 16 |- ( Base ` G ) = ( Base ` G )
33	32	subgss 13380	. . . . . . . . . . . . . . 15 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
34	33	3ad2ant1 1020	. . . . . . . . . . . . . 14 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> S C_ ( Base ` G ) )
35		simp3 1001	. . . . . . . . . . . . . 14 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> X e. S )
36	34, 35	sseldd 3185	. . . . . . . . . . . . 13 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> X e. ( Base ` G ) )
37	36	adantr 276	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> X e. ( Base ` G ) )
38		subgmulgcl.t	. . . . . . . . . . . . 13 |- .x. = (.g ` G )
39		eqid 2196	. . . . . . . . . . . . 13 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
40	32, 7, 38, 39	mulgnn 13332	. . . . . . . . . . . 12 |- ( ( -u N e. NN /\ X e. ( Base ` G ) ) -> ( -u N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) )
41	31, 37, 40	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( -u N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) )
42	35	adantr 276	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> X e. S )
43	38	subgmulgcl 13393	. . . . . . . . . . . 12 |- ( ( S e. (SubGrp ` G ) /\ -u N e. ZZ /\ X e. S ) -> ( -u N .x. X ) e. S )
44	24, 26, 42, 43	syl3anc 1249	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( -u N .x. X ) e. S )
45	41, 44	eqeltrrd 2274	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) e. S )
46		eqid 2196	. . . . . . . . . . 11 |- ( invg ` G ) = ( invg ` G )
47		eqid 2196	. . . . . . . . . . 11 |- ( invg ` H ) = ( invg ` H )
48	1, 46, 47	subginv 13387	. . . . . . . . . 10 |- ( ( S e. (SubGrp ` G ) /\ ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) e. S ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) )
49	24, 45, 48	syl2anc 411	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) )
50	23, 49	syldan 282	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) )
51	13	adantr 276	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) )
52	51	fveq1d 5563	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) = ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) )
53	52	fveq2d 5565	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) )
54	50, 53	eqtrd 2229	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) )
55	54	anassrs 400	. . . . . 6 |- ( ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) )
56		0z 9354	. . . . . . 7 |- 0 e. ZZ
57	19	adantr 276	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> N e. ZZ )
58		zdclt 9420	. . . . . . 7 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> DECID 0 < N )
59	56, 57, 58	sylancr 414	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> DECID 0 < N )
60	55, 59	ifeq2dadc 3593	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) = if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) )
61	16, 60	eqtrd 2229	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) = if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) )
62		0zd 9355	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> 0 e. ZZ )
63		zdceq 9418	. . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
64	19, 62, 63	syl2anc 411	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> DECID N = 0 )
65	61, 64	ifeq2dadc 3593	. . 3 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
66	5, 65	eqtrd 2229	. 2 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
67	32, 7, 2, 46, 38, 39	mulgval 13328	. . 3 |- ( ( N e. ZZ /\ X e. ( Base ` G ) ) -> ( N .x. X ) = if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
68	19, 36, 67	syl2anc 411	. 2 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) = if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
69	1	subgbas 13384	. . . . 5 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
70	69	3ad2ant1 1020	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> S = ( Base ` H ) )
71	35, 70	eleqtrd 2275	. . 3 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> X e. ( Base ` H ) )
72		eqid 2196	. . . 4 |- ( Base ` H ) = ( Base ` H )
73		eqid 2196	. . . 4 |- ( +g ` H ) = ( +g ` H )
74		eqid 2196	. . . 4 |- ( 0g ` H ) = ( 0g ` H )
75		subgmulg.t	. . . 4 |- .xb = (.g ` H )
76		eqid 2196	. . . 4 |- seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) )
77	72, 73, 74, 47, 75, 76	mulgval 13328	. . 3 |- ( ( N e. ZZ /\ X e. ( Base ` H ) ) -> ( N .xb X ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
78	19, 71, 77	syl2anc 411	. 2 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .xb X ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
79	66, 68, 78	3eqtr4d 2239	1 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) = ( N .xb X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 835   \/ w3o 979   /\ w3a 980   = wceq 1364   e. wcel 2167   C_ wss 3157   ifcif 3562   {csn 3623   class class class wbr 4034   X. cxp 4662   ` cfv 5259  (class class class)co 5925   0cc0 7896   1c1 7897   < clt 8078   -ucneg 8215   NNcn 9007   ZZcz 9343   seqcseq 10556   Basecbs 12703   ↾_s cress 12704   +g cplusg 12780   0gc0g 12958   Grpcgrp 13202   invgcminusg 13203  ._gcmg 13325  SubGrpcsubg 13373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-seqfrec 10557  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-mulg 13326  df-subg 13376

This theorem is referenced by:  zringmulg  14230